We will now study the expansion coefficients \(\chi _k\)  of the scattering angleIt is most useful to perform the integral (REF ) directly using the \(p.n\)  gauge
[1]} both in
harmonic and in EOB coordinates. It is advisable to use the necessary integrals from
Ref. [2]}, since not all computer algebra systems do perform them correctly.
\(\frac{\chi ^{\rm loc} + \pi }{2} = \sum _{k=1}^\infty \frac{\chi _k^{\rm loc}}{j^k}= {\sf Reg}_{u_{\rm max}} \int _0^{u_{\rm max}} \frac{j dy}{(p.n)(y,j)},~~ p.n~:=~\sqrt{p_\infty ^2 - j^2 y^2 +W(y,p_\infty )}.\) 
